Universal scaling law for chiral antiferromagnetism

The chiral antiferromagnetic (AFM) materials, which have been widely investigated due to their rich physics, such as non-zero Berry phase and topology, provide a platform for the development of antiferromagnetic spintronics. Here, we find two distinctive anomalous Hall effect (AHE) contributions in the chiral AFM Mn3Pt, originating from a time-reversal symmetry breaking induced intrinsic mechanism and a skew scattering induced topological AHE due to an out-of-plane spin canting with respect to the Kagome plane. We propose a universal AHE scaling law to explain the AHE resistivity (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rho }}_{AH}$$\end{document}ρAH) in this chiral magnet, with both a scalar spin chirality (SSC)-induced skew scattering topological AHE term, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a}_{sk}$$\end{document}ask and non-collinear spin-texture induced intrinsic anomalous Hall term, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{b}}_{{in}}$$\end{document}bin. We found that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{a}}}_{{{sk}}}$$\end{document}ask and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{b}}}_{{{in}}}$$\end{document}bin can be effectively modulated by the interfacial electron scattering, exhibiting a linear relation with the inverse film thickness. Moreover, the scaling law can explain the anomalous Hall effect in various chiral magnets and has far-reaching implications for chiral-based spintronics devices.

3. The author's try to corroborate their findings using DFT calculations and by analyzing the thickness dependence of the observed effect.
Criticism could be made for each of these points.While the quality of the samples seems to be high and the obtained data seems to be valid, the rigor of the data analysis does not live up to the author's claim of discovering a "universal" scaling law.For the following reasons: it is very difficult to judge the quality of the supposed law solely based on Figure 3.No error estimates are made and the relevant data series for Mn3Pt is tugged away in the corner where it is difficult to evaluate how closely the scaling law is actually traced by the experimental data.I would encourage the author's to improve this part of the presentation, since it is one of the core results of the paper.
Concerning the thickness dependence, it did not became very clear to me how the magnetic structure behaves as a function of the film thickness and how this correlates with the scalar spin chirality.For example, Figure 4 b) shows a schematic of how the magnetic structure might evolve.The problem is: in the 3-site model, the SSC is zero for both the purely ferromagnetic configuration and the purely antiferromagnetic configuration.It attains a maximum at intermediate canting angles.One would therefore expect a non-monotonous behavior of the SSC if the evolution of the magnetic texture would be as drastic as shown in Figure 4 b).I see this as another critical piece of the author's argument and would encourage them to lay out their argument more carefully with a special scrutinty on the evolution of the magnetic texture as a function of the film thickness.
And with regard to the DFT calculations it would be helpful to specify more transparently, how the magnetic texture was set up.I assume it was a purely antiferromagnetic state, which was then slowly canted towards 111-direction?
Overall, I believe that the results of the manuscript are potentially interesting and relevant to the field.But the quality of the presentation currently does not meet the standard of the journal and needs to be improved as it was layed out above.As of now, I would therefore not recommend the publication until those points have been adressed.
Reviewer #3: Remarks to the Author: In the manuscript entitled 'Universal scaling law for chiral antiferromagnet', by Shijie Xu et al., submitted to Nature Communications, the authors have studied the anomalous Hall effect (AHE) of the chiral non-collinear antiferromagnet (AFM) Mn3Pt films grown on MgO(001) substrate.The AHE, they observed, has two distinctive contributions; one is an intrinsic mechanism induced by a time-reversal symmetry breaking, and the second is a skew scattering induced by topological AHE due to a scalar spin chirality.To distinguish these two AHE contributions, they used a universal AHE scaling law, which was previously proposed to explain the AHE resistivity (rho_AHE) in Mn3Pt: rho_AHE = a_sk* rho_xx+b_in*rho_xx^2 where rho_xx is longitudinal resistivity and a_sk (b_in) is the skew scattering topological AHE (the intrinsic AHE).The main finding is that the both a_sk and b_in parameters change with the reversal film thickness d.They explained this behavior by considering the enhancement/reduction of the out-of-plane anisotropy due to spin canting, which is also supported by band calculations.
In my opinion, their finding and their conclusions are insufficient for publication in Nature Communications.Since they used the scaling law already proposed, the title of the manuscript "universal scaling law for chiral antiferromagnet" sounds very exaggerated.
There should be several steps and let me elaborate on these points below: (1) The scaling is conducted by rho_AHE and rho_xx obtained around room temperature, which is unusual.In line number 76, the authors wrote "the new scaling law is universal for describing the AHE of chiral magnet".I could not find the "new scaling law" in the manuscript.
(2) The authors tried to elucidate the origin of the thickness dependence of AHE by using magnetic anisotropy.But this argument should rely on experimental evidence, ex magnetic properties.In Fig1, they presented an MH curve but only one thickness and only a c-axis magnetic field.Please present additional data of different thicknesses and different magnetic field axis (ex.In-plane).Otherwise, I can not believe their conclusion about the thickness dependence.
(3) The scaling law of AHE was proposed for zero temperature in theory.So the conclusions made only with room temperature measurements are not appropriate.
(4) In Fig. 4, the authors present a_sk and b_in as a function of 1/d.How did they get them?Please put an error bar here because their thickness dependencies are apparently small.
(5) Shape of the Ryz-H curves presented in Fig. 2 is totally different from that of the M-H curve in Fig. 1.Please explain the reason.
(6) Did the author include spin-orbit coupling (SOC) in the band calculation?I guess that it might give a non-negligible effect because Pt has large SOC.At least, please mention with or without SOC in the calculation setup.Response 2: Thank reviewer 2 very much for your suggestion.We revised the figure 4 and show the measured magnetizations and SSC increased at larger film thickness.Based on our experiment data and DFT calculation, the SSC will increase at a smaller range and approach to the maximum value.
And with regard to the DFT calculations it would be helpful to specify more transparently, how the magnetic texture was set up.I assume it was a purely antiferromagnetic state, which was then slowly canted towards 111-direction?
Response 3: Thank reviewer 2 very much for your suggestion.We revised the figured 5 and shown how the magnetic texture was set up.The magnetic moment is initially set to a purely antiferromagnetic state, then we slightly tilt the magnetization of Mn atoms, the total net magnetization is set to the [111] direction of the Mn3Pt lattice.And set up is shown in lineV252 to 254 of the new manuscript.1.A separate 'response to referees' letter that addresses the referees 3' in a point-by-point manner

Response：
In my opinion, their finding and their conclusions are insufficient for publication in Nature Communications.Since they used the scaling law already proposed, the title of the manuscript "universal scaling law for chiral antiferromagnet" sounds very exaggerated.
There should be several steps and let me elaborate on these points below: (1) The scaling is conducted by rho_AHE and rho_xx obtained around room temperature, which is unusual.In line number 76, the authors wrote "the new scaling law is universal for describing the AHE of chiral magnet".I could not find the "new scaling law" in the manuscript.
Response 1a: Thank reviewer 3 very much for your suggestion.Your suggestion makes this article more scientific and rigorous.For sure, ρAHE and ρxx obtained around zero temperature is better to conduct the scaling Law.However, it is hard to obtain the ρAHE below room temperature for Mn3Pt due to larger saturation field at low temperature as shown at figure Ⅰ.On the other hand, previous paper show that the scaling around room temperature and low temperature is the same and both of them is linear ( 1, linear scaling law at 5-320 K ,Physical review letters 103.8 (2009): 087206.2,linear scaling law at 5-300 K , Physical Review Letters 109.6 (2012): 066402 ).Finally, our experiment data also show the linear scaling law at 300-390 K, it should also be rigorous.(2) The authors tried to elucidate the origin of the thickness dependence of AHE by using magnetic anisotropy.But this argument should rely on experimental evidence, ex magnetic properties.In Fig1, they presented an MH curve but only one thickness and only a c-axis magnetic field.Please present additional data of different thicknesses and different magnetic field axis (ex.In-plane).Otherwise, I can not believe their conclusion about the thickness dependence.

Response 2：
Thank Referee 3 for his/her useful advice.Your suggestion makes this article more rigorous.We have added experiment data of different thicknesses as shown in figure II and revised the Fig. 4b.(3) The scaling law of AHE was proposed for zero temperature in theory.So the conclusions made only with room temperature measurements are not appropriate.(5)Shape of the Ryz-H curves presented in Fig. 2 is totally different from that of the M-H curve in Fig. 1.Please explain the reason.
Response 5: Thank Referee 3 for his/her useful advice.The M-H loop reflects the evolution of the net magnetic moment.The Ryz-H loop describes the evolution of the Berry curvature, which does not necessarily follow the net moment of the antiferromagnet.Also, in our article, we mention " It is worth noting that the magnetic coercivity is much smaller than that of AHE coercivity, because the Zero magnetic moment indicates that the spin canting is zero.However, the AHE coercivity indicates that the topological AHE and the intrinsic AHE have opposite and equal contributions." (6) Did the author include spin-orbit coupling (SOC) in the band calculation?I guess that it might give a non-negligible effect because Pt has large SOC.At least, please mention with or without SOC in the calculation setup.
Response 6: Thank Referee 3 for his/her useful advice.Sure, we have considered the spin-orbit coupling (SOC) in the band calculation and the details can be found in the methods section.
Response 7: Thank Referee 3 for his/her useful advice.we revised the Fig. 2 and the article.

Reviewers' Comments:
Reviewer #3: Remarks to the Author: I have carefully read the revised manuscript and the author's response.I found some of their responses and manuscript revisions confusing.For instance, in response 2 to one of my comments regarding the lack of experimental evidence for thickness-dependent magnetic anisotropy, the authors presented a dataset.However, they did not provide a detailed explanation of how they reached their conclusion about the magnetic anisotropy, as depicted in Figure 4b.Additionally, in the main text (line 153), it is mentioned that the perpendicular magnetic anisotropy in the Mn3Pt films changes with different thicknesses (Fig. 2b).However, I could not find this data in the manuscript, or at least, it was not apparent to me.These points are crucial for validating the new scaling law they claim, as they are connected to the scalar spin chirality that induces the topological anomalous Hall effect, which is the first term of their scaling law.Therefore, I kindly request a detailed explanation of the interpretation of the MH curve and how it relates to the evolution of scalar spin chirality as a function of thickness.Without this clarification, I believe that the work is not suitable for publication in Nature Communications.
Reviewer #4: Remarks to the Author: In my opinion, the authors have not sufficiently addressed the concerns raised in the previous round of review.This concerns mostly the following points: 1.The quality of the presentation of the data and the data analysis itself does not live up to the claim of a discovered "universal scaling law".It is still very difficult to judge whether or nor the data actually supports their claim.Among other things this is made difficult also by the logarithmic scaling of the x-axis in Figure 3 and the barely visible data series.2. At the same time, the scaling relation would not be an entirely new insight since for example reference [9] in the revised manuscript clearly describes skew scattering contributions to the AHE via the scalar spin chirality.
Based on this, I cannot recommend the publication in Nature Communications.
Reviewer #3 (Remarks to the Author): I have carefully read the revised manuscript and the author's response.I found some of their responses and manuscript revisions confusing.For instance, in response 2 to one of my comments regarding the lack of experimental evidence for thickness-dependent magnetic anisotropy, the authors presented a dataset.However, they did not provide a detailed explanation of how they reached their conclusion about the magnetic anisotropy, Additionally, in the main text (line 153), it is mentioned that the perpendicular magnetic anisotropy in the Mn3Pt films changes with different thicknesses (Fig. 2b).However, I could not find this data in the manuscript, or at least, it was not apparent to me.These points are crucial for validating the new scaling law they claim, as they are connected to the scalar spin chirality that induces the topological anomalous Hall effect, which is the first term of their scaling law.Therefore, I kindly request a detailed explanation of the interpretation of the MH curve and how it relates to the evolution of scalar spin chirality as a function of thickness.Without this clarification, I believe that the work is not suitable for publication in Nature Communications.
Point one：in response 2 to one of my comments regarding the lack of experimental evidence for thickness-dependent magnetic anisotropy Point two：perpendicular magnetic anisotropy in the Mn3Pt films changes with different thicknesses (Fig. 2b) Point three：A detailed explanation of the interpretation of the MH curve and how it relates to the evolution of scalar spin chirality as a function of thickness Thank Reviewer #3 for your suggestion.We have not only measured a large number of anomalous Hall data, but also supplemented magnetic measurements，even though the antiferromagnetic magnetic moment is very difficult to measure and very small.
For the point one about the antiferromagnetic magnetic anisotropy, few experimental groups have reported it.the magnetic anisotropy energy (due to spin canting effect) can be described as K A = |∫  hard axis *  − ∫  easy axis * |. is the magnetic field,  is the net moment.However, H. Chen et al. (Phys. Rev. Lett. 112, 017205) theoretically predicted the evolution of the magnetic structure about Mn3Ir (which have the same structure of Mn3Pt).The antiferromagnetic spin structure will tilt upward under positive saturation magnetic field and have the opposite result under negative magnetic field.When the large magnetic field was applied at Hard axis, the net moment go to zero ∫  hard axis *  = 0, So the K A = ∫  easy axis * .Therefore, the perpendicular magnetic anisotropy energy in the Mn3Pt films with different thicknesses can described at figure 1.The magnetizations M induced by the net moment increases with the thickness, which means that the spin structure will tilt larger with the thickness.At a result, the scalar spin chirality will increase at larger thickness, causing the larger topological anomalous Hall effect.
(7) I cannot find Fig.2e and Fig.2f. 1.A separate 'response to referees' letter that addresses the referees 2' in a point-by-point manner Response： Reviewer #2 (Remarks to the Author): Criticism could be made for each of these points.While the quality of the samples seems to be high and the obtained data seems to be valid, the rigor of the data analysis does not live up to the author's claim of discovering a "universal" scaling law.For the following reasons: it is very difficult to judge the quality of the supposed law solely based on Figure 3.No error estimates are made and the relevant data series for Mn3Pt is tugged away in the corner where it is difficult to evaluate how closely the scaling law is actually traced by the experimental data.I would encourage the author's to improve this part of the presentation, since it is one of the core results of the paper.Response 1: Thank reviewer 2 very much for your suggestion.Your suggestion makes this article more scientific and rigorous.We added error bar at figure4 and revised figure3 according to your suggestion.

Figure 5 .
Figure 5. (a-b) Band structure of R-Γ path for Mn3Pt, the color represent the spin polarization component on [111] direction.(c-d) Band structure near Fermi energy (upper panel) and Berry curvature (lower panel) in atomic units along the symmetry lines.The black line represents the purely antiferromagnetic state, the red line represents the state with net magnetization along [111] direction, whose z-component is 0.30μ_B.(e) Calculated intrinsic contribution to the AHC versus Fermi energy with different net magnetizations.(f) Calculated intrinsic contribution to the AHC versus different total magnetizations at the Fermi level.

Figure Ⅰ ,
Figure Ⅰ, Magnetic-filed-dependent anomalous Hall resistance at 30 nm for chiral magnets Mn3Pt, the measured temperature is 250 K.

Figure
Figure ⅠI,Magnetic-filed-dependent magnetization data at different thickness for chiral magnets Mn3Pt, the measured temperature is 300 K.
Thank Referee 3 for his/her useful advice.We get the a and b by the Linear scaling fitting( = + ) at figure 3. We revise the figure 4 and add the error bar.